Leonard systems are related to Leonard pairs as follows.
For the Leonard system \(\Phi \)  from (REF ),
by [1]} the pair \(A,A^*\)  is a Leonard pair on \(V\) .
Conversely, for a Leonard pair \(A,A^*\)  on \(V\) , each of \(A,A^*\)  is multiplicity-free
by Lemma REF .
Moreover there exists an ordering \(\lbrace E_i\rbrace _{i=0}^d\)  of the primitive idempotents of \(A\) 
and an ordering of \(\lbrace E^*_i\rbrace _{i=0}^d\)  of the primitive idempotents of \(A^*\) 
such that \((A; \lbrace E_i\rbrace _{i=0}^d; A^*; \lbrace E^*_i\rbrace _{i=0}^d)\)  is a Leonard system on \(V\) 
(see [1]}.)
